US9661084B2 - Method and system for sampling online communication networks - Google Patents
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- US9661084B2 US9661084B2 US14/039,086 US201314039086A US9661084B2 US 9661084 B2 US9661084 B2 US 9661084B2 US 201314039086 A US201314039086 A US 201314039086A US 9661084 B2 US9661084 B2 US 9661084B2
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L67/00—Network arrangements or protocols for supporting network services or applications
- H04L67/50—Network services
- H04L67/51—Discovery or management thereof, e.g. service location protocol [SLP] or web services
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- G—PHYSICS
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- G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
- G06Q50/00—Systems or methods specially adapted for specific business sectors, e.g. utilities or tourism
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L67/00—Network arrangements or protocols for supporting network services or applications
- H04L67/50—Network services
- H04L67/535—Tracking the activity of the user
Definitions
- the invention relates generally to the field of social networking and more particularly to generating representative sampling from a social network dataset.
- Analyzing or mining online social networks has become one of the most pressing problems of modern-day data mining. The need arises due to the exponential growth of these networks as they become increasingly popular.
- Networks such as Facebook®, Twitter®, and LinkedIn® include vast amounts of information and vast amounts of interconnection information that is useful for many purposes. Some non-limiting examples of purposes include commercial purposes, management purposes, political purposes, research purposes, demographic purposes, emergency preparedness applications, and defense and security applications.
- sample data is preferably representative of the whole data set or, alternatively the results of the analysis and sampling are together representative of the dataset.
- sampling is achievable by crawling online social networks to find a relatively small representative sample suitable, for example, for studying properties and testing algorithms. The sample is extracted through the crawling and then used for more responsive data analysis.
- a number of existing techniques for crawling include the breadth-first search (BFS) and random walks (RW). It is known that these techniques usually yield a bias toward the most highly connected nodes. With social networks, this is highly problematic as some nodes have such high connectivity—imagine someone famous—that they skew resulting samples. That said, crawling using the traditional Metropolis-Hasting algorithm (MH), which is a typical Monte Carlo Markov Chain (MCMC) technique, can create unbiased samples suitable for the problem of social network analysis and social network activity analysis.
- MH Metropolis-Hasting algorithm
- MCMC Monte Carlo Markov Chain
- the breadth-first search (BFS) method which is regarded as a graph traversal technique, explores the next node assuming the traditional breadth-first search algorithm. It has been used practically for sampling online social networks in past research. Recent research also shows that the methodology sometimes densely covers a specific region of a graph due to incomplete search, but this bias is potentially correctable by deriving an unbiased estimator of the original node degree distribution. That said, even such an estimator may be difficult to derive.
- the random walk (RW) method chooses a next state W uniformly and at random among the neighbors of a current node V. Because the probability of the RW at the particular node V converges, the random walk sample nodes are biased towards high degree nodes. This bias may be corrected by an appropriate re-weighting of the measured value such as the Hansen-Hurwitz estimator. That said, the biasing is problematic in most social networks as some nodes are extremely high degree relative to others.
- the Metropolis-Hasting Random Walk (MHRW) method appropriately modifies transition probabilities so that over sufficient time it converges to a uniform distribution.
- the Metropolis-Hasting process is a typical Markov Chain Monte Carlo (MCMC) technique for sampling from a probability distribution.
- MCMC techniques such as the Metropolis-Hasting process come with significant challenges: significant burn-in lengths and correlation with initial node choice are just two significant drawbacks. This usually leads to slow mixing.
- MHRW Random Walk
- various convergence diagnostic methods such as Geweke Diagnostic cannot guarantee the chain has converged to a sample value from the desired distribution. Therefore, the sample, e.g., 78 k, obtained by such MCMC algorithm is usually approximate. Verification of the sample is a time consuming task because the data set, which is very large, must be analysed to determine the representative nature of the sample.
- a method of providing an online social network comprising at least a data store for storing data relating to a plurality of connections forming a state space and a communication port for supporting communication with individuals to whom the connections relate, the individuals communicating with the online social network via a wide area communication network; providing a first seed value; providing a coupling-from-the-past process having an update function for resulting in a non-trivial state space smaller than the state space of the online social network and forming a representative sample thereof; based on the first seed value selecting at least a first node; retrieving from the online social network dataset via the wide area communication network data based on the selected at least a first node; and applying the coupling-from-the-past process having the update function to the at least a first node to determine a non-trivial state space based on the first seed value, the non-trivial state space smaller than the state space of the online social network and the non-trivial state space
- a method of sampling an online social network dataset comprising: providing a first seed value; providing a coupling-from-the-past process having an update function for resulting in a non-trivial state space smaller than the state space of the online social network and forming a representative sample thereof; based on the first seed value selecting at least a first node; retrieving from the online social network dataset via a computer communication network data based on the selected at least a first node; and applying the coupling-from-the-past function having the update function to the at least a first node to determine a first non-trivial state space based on the first seed value, the first non-trivial state space smaller than the state space of the online social network and the non-trivial state space forming an intermediate state in determining a representative sample of the online social network.
- a method comprising of deterministically determining a representative sample of a large online graph by selecting at least a first node and iterating until a process coalesces on the representative sample.
- a method sampling an online social network dataset comprising: providing a statistical description of a representative sample of a state space; determining based on the statistical description a minimum number of nodes within a representative sample meeting the statistical description; performing at least two separate processes on a same online social network dataset to determine at least two representative samples each having at least the minimum number of nodes therein for being combined into a single larger representative sample.
- a method of sampling an online social network dataset comprising: providing a statistical description of a representative sample of a state space; determining based on the statistical description a first number of nodes within a representative sample meeting the statistical description; at intervals automatically extracting a representative sample having the first number of nodes therein from an online social network dataset.
- FIG. 1 is a sample graph with weighted interconnections
- FIG. 2 is a simplified node diagram showing node traversal for identifying a trivial state space using the NTSS with the example of FIG. 1 ;
- FIG. 3 is a simplified block diagram of a first user system
- FIG. 4 is a simplified block diagram of a network including a social network.
- FIG. 5 is a simplified diagram of a plurality of nodes in a social network.
- a computer 31 is coupled to a display 32 , a mouse 33 and a keyboard 34 .
- the entire computer system is integrated, for example within a laptop, tablet or smart phone.
- FIG. 4 shown is a simplified network diagram showing a network 45 in the form of the Internet.
- a computer 42 similar to that of FIG. 3 is shown coupled to the Internet.
- server 44 Also coupled to the Internet are server 44 , computer 43 and online social network server and data storage 41 .
- node 51 is connected to nearly every other node.
- Nodes 52 and 53 are not connected to node 51 .
- This node 51 has a high order of connectivity. If the nodes are traversed at random, it can easily be seen that node 51 would be traversed more than node 52 .
- Nodes 54 , 55 , 56 , and 57 form a tight knit social group. As opposed to other groups of nodes shown with looser social coupling. Clearly, it is difficult to determine a statistically relevant representative sample, even of this very small graph.
- the Facebook® social network graph comprises hundreds of millions of nodes.
- a coupling-from-the-past process allows for perfect sampling from a given distribution.
- a quality of sampling is dependent upon many factors. That said, once the parameters and quality of sampling are determined based on the update function and the process itself, representative samples are automatically generatable by the process repeatedly and at different times.
- the update function once defined allows for automated sample generation from the ever-changing data sets that make up social networks.
- such a system allows for generation of a sample when needed that is representative of a much larger data set and that can be analysed further and with less difficulty than the larger data set.
- P be a transition probability defining an ergodic Markov chain on state space ⁇ .
- the composite map F t1 t2 which describes evolution of Markov chains from t 1 to t 2 given any initial state x, is defined as
- F 0 t and F ⁇ t 0 define forward coupling and the backward coupling Markov chains, respectively.
- the forward coupling Markov chains do not necessarily produce a sample from a stationary distribution and are subject to a bias due to changed coalescence time.
- backward coupling produces a “perfect” sample within limits without any bias due to a fixed time for coalescence detection.
- 1 end
- the update function ⁇ t for evolution of Markov chains when only a local transition probability P is available.
- the update function is also termed a random map, which is just a random function representation of P as described above.
- An invalid ⁇ t sometimes leads to failure of coalescence in coupling-from-the-past process.
- the update function maps a set of nodes X t to a new set of nodes X t+1 .
- Each node in X t is mapped to a new adjacent node based on a probability of transition.
- the probability of transition is typically a global property. Which adjacent node to select is determined by random parameter R t+1 .
- the update function ⁇ can be written in terms of a range (R lower ,R upper ] for which a transition occurs.
- a probability of mapping x to one of its neighbors using the update function is equal to the transition probability.
- the key property of the update function ⁇ is that it is deterministic in R t+1 . Given a set of initial states X ⁇ T and a Markov Chain R ⁇ T , R ⁇ T+1 , . . . , R ⁇ 1 , R 0 the set of states, nodes, and paths is deterministic.
- ⁇ ⁇ ( x i , R t + 1 ) x i , j , ⁇ ⁇ if ⁇ ⁇ R t + 1 ⁇ ( j k i , j + 1 k i ] ( 7 )
- FIG. 1 shown is a simplified diagram of a toy example from the literature.
- the toy example shows a directed graph network. Instead of estimating transition probabilities using degrees of a node and nodes adjacent thereto, the toy example typically has defined global transition probabilities of a chain. Therefore, given a current state 1, the update function based on Metropolis-Hasting process and a coupling-from-the-past process is actually estimated by using equation 10.
- the update function defines a random map, and is a deterministic function, ⁇ ( R t ): ⁇ (11)
- Some states in ⁇ might not be necessary to be start states of Markov chains. Typically, a small ⁇ ′ ⁇ is sufficient to be start states for global coupling. A small and non-trivial subset ⁇ ′ is much desired in practice instead of whole state space ⁇ . In particular, sampling on a large online social network using coupling-from-the-past to produce a representative sample within known limits or even to produce a near perfect sample would be advantageous.
- ⁇ (R t ) and F t1 t2 (x) define a new random map as ⁇ ( R t ) or F t1 t2 : ⁇ ′ approaches ⁇ (12)
- a non-trivial state space ⁇ ′ is a subset of the state space ⁇ . Simply, ⁇ is also a non-trivial state space by itself.
- F ⁇ M 0 ( ⁇ ′) coalesces to an exact sample from ⁇ given a M. This is justified because F ⁇ M 0 ( ⁇ ′) has the same distribution as ⁇ .
- NTSS non-trivial state space process
- a Non-Trivial State Space process is set out in the following:
- the NTSS produces a non-trivial state space as shown in FIG. 2 .
- t ⁇ 1,
- ⁇ 0, 1, 2 ⁇ .
- the online coupling-from-the-past is used for approximately perfect sampling of online social networks.
- the process first generates a non-trivial state space ⁇ ′ from given initial states X 0 using the proposed NTSS.
- Independent perfect samples are obtained by independently executing a standard coupling-from-the-past with the customized update function discussed herein and the obtained non-trivial state space ⁇ ′.
- other updates functions are also supported so long as they achieve coalesced approximately perfect representative samples. More details about the online coupling-from-the-past is set out as follows:
- the uniform method allows for a Markov chain evolving from a current state to a next new state with equal probability.
- the Metropolis-Hasting method heuristically estimates a probability distribution for transition of states in a Markov chain by estimating local density of nodes. Therefore, the Metropolis-Hasting process is often more powerful than a uniform method when applied to complex networks.
- the update function is set by selecting either the RW or the MH methods for calculating transition probabilities; further, we show that the update function with the MH method is more robust than the update function with the RW method for sampling on complex networks.
- a subset of the whole state space, called non-trivial state space can be identified by using the proposed Non-Trivial State Space algorithm, which is implemented by assuming a novel strategy to modify the standard coupling-from-the-past. Finally, a coupling-from-the-past process for sampling online social networks is described.
- the proposed update function is a function of a random variable and is governed by transition probability of a node.
- the update function is dominated by random numbers. Therefore, the update function is also regarded as a time-related random map, and its probability is equal to a transition probability.
- R t the random number is large, e.g., R t ⁇ 1. This is regarded as an exceptional transition for evolution of Markov chains. Avoiding this exceptional transition is beneficial for producing coalescence in the online coupling-from-the-past.
- Rt ⁇ R t + R ⁇ i , if ⁇ ⁇ R t + R ⁇ i ⁇ 1 ⁇ R t + R ⁇ i - 1 ⁇ , otherwise ( 14 ) Because R t is a uniform random number and ⁇ circumflex over (R) ⁇ i is fixed for each x i , R i t is also uniformly distributed in (0, 1]. The equation is only to translate the original R t in (0, 1]. This avoids self-transitions occurring at the same time steps in all Markov chains.
- Another of other methods of selecting nodes based on random numbers is employed. Many such methods likely exist. For example, another way to select nodes based on random values is by binning nodes like so:
- Can1 Can 2 CurrNode Can3 CurrNode Can4 (0 . . . (0.25 . . . (0.375 . . . (0.5 . . . (0.5625 . . . (0.75 . . . 1) 0.25) 0.375) 0.5) 0.5625) 0.75)
- a result of the coupling-from-the-past process is or approximates a perfect sampling from a given space, then it is possible to execute numerous coupling-from-the-past processes, each with different seed values, to generate a plurality of independent approximately perfect samples. Combining of the approximately perfect samples results in a larger sample that is also approximately perfect. So long as there are no resulting duplicate nodes within the larger sampling, the process—executed in parallel for example—avoids oversampling.
- the term perfect as used here is theoretical and in practice some level of quantization and noise is present in an output sample from the process. Results and samples are merely within limits of the perfect sample.
- a coupling-from-the-past process is executed on the Twitter® network to extract a representative sample of 100 nodes. A sample of 1000 nodes is sought, so the process is executed 10 times with 10 different seed values. The 1000 resulting nodes are amalgamated into the final output sample. Other than oversampling—a single node being within the output sample more than one time—the result is representative of the Twitter® network. Alternatively, the coupling-from-the-past process is executed 20 times and those resulting samples that have no overlapping nodes are selected such that 1000 unique nodes are present within the 1000 node output result.
- the error present in the output sample relative to a perfect sample is quantifiable statistically at the outset.
- a definition of sampling would give a level of accuracy and a likelihood of accuracy such as 99% 19 out of 20 times—and the acceptable error is determinative of the minimum sample size for each coupling-from-the-past process—here 100 nodes—and the overall criteria for sample generation. That said, because of its deterministic nature, the coupling-from-the-past process is very well suited to generating samples within statistical limits as pre-defined.
- perfect is used herein, it is not intended to refer to a theoretically perfect sample, but to a sample meeting statistical requirements within predetermined statistical limits.
Abstract
Description
Pr(Φ(x)=y)=P(x,y) (3)
-
- Input M: initial time (>0), Ω: state space
- Output X0: singleton state
begin | ||
T = M, T0 = 0 | ||
repeat | ||
R−T+1, R−T+2, . . . , RTo ~U(0,1) | ||
X−T = Ω, t=−T | ||
while t < 0 | ||
XT+1 = Φ(Xt,Rt+1) | ||
t = t + 1 | ||
T0 = −T, T = 2T | ||
until |X0| = 1 | ||
end | ||
X t+1=Φ(X t ,R t+1) (5)
where Rt+1˜U(0, 1) and Xt, Xt+1 ⊂Ω; this defines a random map as Φt(χ):Ω approaches Ω, χεΩ.
and the update function is given by
where ki,j is the degree of the adjacent node xi,j. The Metropolis-Hasting process allows for self transition with probability
P(x i ,x i,j)=1−Σj P(x i ,x i,j) (9)
Then the update function is given by
Φ(R t):Ω→Ω (11)
Φ(R t) or F t1 t2:Ω′ approaches Ω (12)
-
- Given Φ(Ω, Rt), denoted as Φt(Ω), and the composite function
- Ft1 t2(χ)=(Φt2-1·Φt2-2 . . . Φt1)(χ), where t1<t2 and ∀χεΩ, if the time distance |t1−t2| is sufficiently large, s.t., Ft1 t2(χ)⊂Ω′⊂Ω, then Ω′ is called a non-trivial state space with respect to t2.
-
- Input N: the size of non-trivial state space, X0: initial states τ: the coefficient of non-trivial states, e.g., for a large network;
- Output: a non-trivial state space
begin | ||
s = 0, Ω{grave over ( )} = X0 | ||
while |Ω{grave over ( )}| < N or |Ω{grave over ( )}| > s | ||
i = 0, T = −1, T0 = 0 | ||
repeat | ||
t = T, Xt = Ω{grave over ( )}, s=|Ω{grave over ( )}| | ||
RT+1, . . . , RTo~U(0,1) | ||
while t < 0 | ||
Xt+1 = Φ(Xt, Rt+1) | ||
t = t + 1 | ||
T0 = T, T = 2T | ||
i = i + 1 | ||
until i ≧ τ V |Xt| = 1 | ||
Ft1 t2(χ) ⊂ Ω{grave over ( )} = Ω{grave over ( )} ∪ Xt | ||
end | ||
-
- Input n: the total number of independent perfect samples, X0: initial states consisting of some nodes as sample seeds from a given online social networks, e.g., Facebook, etc;
- Output S: a collection of independent perfect samples
begin | ||
Ω{grave over ( )}= NTSS(n, X0), | ||
i = 0, T = 1, S = { } | ||
while i < n | ||
s = coupling-from-the-past(T, Ω{grave over ( )} ), by φ in (7) or (10) and (14) | ||
i = i + 1 | ||
S = S ∪ {s} | ||
end | ||
Because Rt is a uniform random number and {circumflex over (R)}i is fixed for each xi, Ri t is also uniformly distributed in (0, 1]. The equation is only to translate the original Rt in (0, 1]. This avoids self-transitions occurring at the same time steps in all Markov chains.
Can1 | Can 2 | CurrNode | Can3 | CurrNode | Can4 |
(0 . . . | (0.25 . . . | (0.375 . . . | (0.5 . . . | (0.5625 . . . | (0.75 . . . 1) |
0.25) | 0.375) | 0.5) | 0.5625) | 0.75) | |
(u*CurrNode) % 1<=min(1,CurrNode/CanNode)
This makes the MH selection efficient and deterministic
Claims (23)
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